Standing Waves for Discrete Nonlinear Schr?dinger Equations with Nonperiodic Bounded Potentials
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摘要
In this paper, we investigate standing waves in discrete nonlinear Schr?dinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we prove the existence and infinitely many sign-changing solutions of the equation. The results on the exponential decay of standing waves are also provided.
In this paper, we investigate standing waves in discrete nonlinear Schr?dinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we prove the existence and infinitely many sign-changing solutions of the equation. The results on the exponential decay of standing waves are also provided.
In this paper, we investigate standing waves in discrete nonlinear Schr?dinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we prove the existence and infinitely many sign-changing solutions of the equation. The results on the exponential decay of standing waves are also provided.
引文
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[23]Zhang,G.Breather solutions of the discrete nonlinear Schr¨oinger equations with unbounded potentials.J.Math.Phys.,50:013505(2009)
[24]Zhang,G.,Pankov,A.Standing waves of the discrete nonlinear Schr¨oinger equations with unbounded potentials,II.Appl.Anal.,89:1541-1557(2010)
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[3]Eilbeck,J.C.,Johansson,M.The discrete nonlinear Schr¨odinger equation:20 years on.In:Localization and Energy Transfer in Nonlinear Systems,edited by L.Vasquez,R.S.MacKay,M.P.Zorzano,World Scientific,Singapore,2003,44-67
[4]Flach,S.,Gorbach,A.Discrete breathers-advances in theory and applications.Phys.Rep.,467:1-116(2008)
[5]Hardy,G.H.,Littlewood,J.E.,P′olya,G.Inequalities,2nd Ed.,Cambridge University Press,Cambridge,1952
[6]Kato,T.Perturbation theory for linear operators.Springer-Verlag,?,1966
[7]Kevrekides,P.G.,Rasmussen,K.?,Bishop,A.R.The discrete nonlinear Schr¨oinger equation:A survey of recent results.Int.J.Modern Phys.B,15:2833-2900(2001)
[8]Kopidakis,G.,Aubry,S.,Tsironis,G.P.Targeted energy transfer through discrete breathers in nonlinear systems.Phys.Rev.Lett.,87:165501(2001)
[9]Liu,Z.L.,Su,J.B.,Weth,T.Compactness results for Schr¨oinger equations with asymptotically linear terms.J.Differential Equations,231:501-512(2006)
[10]Liu,Z.L.,Sun,J.X.Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations.J.Differential Equations,172:257-299(2001)
[11]Livi,R.,Franzosi,R.,Oppo,G.L.Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation.Phys.Rev.Lett.,97:060401(2006)
[12]Pankov,A.Gap solitons in periodic discrete nonlinear Schr¨oinger equations.Nonlinearity,19:27-40(2006)
[13]Pankov,A.Gap solitons in periodic discrete nonlinear Schr¨oinger equations,II:A generalized Nehari manifold approach.Discrete Contin.Dyn.Syst.Ser.A,19:419-430(2007)
[14]Pankov,A.,Rothos,V.Periodic and decaying solutions in discrete nonlinear Schr¨oinger equation with saturable nonlinearity.Proc.Roy.Soc.Ser.A,464:3219-3236(2008)
[15]Pankov,A.Standing waves for discrete nonlinear Schr¨oinger equations:sign-changing nonlinearities.Appl.Anal.,92:308-317(2013)
[16]Schechter,M.,Zou,W.Sign-changing critical points from linking type theorems.Trans.Amer.Math.Soc.,358:5293-5318(2006)
[17]Shi,H.P.,Zhang,H.Q.Existence of gap solitons in periodic discrete nonlinear Schr¨oinger equations.J.Math.Anal.Appl.,361:411-419(2010)
[18]Sun,J.X.Nonlinear functional analysis and its applications.Science Press,Beijing,2008(in Chinese)
[19]Teschl,G.Jacobi operators and completely integrable nonlinear lattices.Amer.Math.Soc.,Providence,RI,New York,2000
[20]Weidmann,J.Linear operators in Hilbert spaces.Springer-Verlag,New York,1980
[21]Yang,M.B.,Chen,W.X.,Ding,Y.H.Solutions for discrete periodic Schr¨oinger equations with spectrum 0.Acta Appl.Math.,110:1475-1488(2010)
[22]Zhang,G.Breather solutions of the discrete nonlinear Schr¨oinger equations with sign changing nonlinearity.J.Math.Phys.,52:043516(2011)
[23]Zhang,G.Breather solutions of the discrete nonlinear Schr¨oinger equations with unbounded potentials.J.Math.Phys.,50:013505(2009)
[24]Zhang,G.,Pankov,A.Standing waves of the discrete nonlinear Schr¨oinger equations with unbounded potentials,II.Appl.Anal.,89:1541-1557(2010)
[25]Zhou,Z.,Yu,J.S.On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems.J.Differential Equations,249:1199-1212(2010)